<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">G1</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font>;<br/>







<b>assume </b><a NAME="E1:76"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">G1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v1</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 ;<br/>

<a NAME="E2:76"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">G1</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a> 
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G1A</font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G1</font>,<font color="Maroon">v1</font>;<br/>
<a NAME="E4:76"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G2</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a>  <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">G1A</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:76"/><i><font color="Green">E44</font></i>: 
not <font color="Maroon">G1</font> is <a href="glib_000.html#V4">trivial</a>
 
<b>by </b><i>, <a class="txt" href="glib_002.html#E5:76"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E59</font></i>:  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G1A</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v2</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, <a class="ref" href="glib_000.html#T50">GLIB_000:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v2</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
;<br/>
<i><font color="Green">E60</font></i>:  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G1A</font> = 
<font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_000.html#T50">GLIB_000:50</a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
<br/>.= 
<font color="Maroon">G2</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>
;<br/>
<b>let </b><font color="Maroon">G2A</font> be   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G2</font>,<font color="Maroon">v2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="glib_002.html#D10" target="_self">GLIB_002:def 10</a><br/>

<a NAME="E8:76"/><i><font color="Green">E61</font></i>: 
not <font color="Maroon">G2</font> is <a href="glib_000.html#V4">trivial</a>
 
<b>by </b><i>, , <a class="ref" href="glib_000.html#T92">GLIB_000:92</a></i>;<br/>
<a NAME="E9:76"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2A</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G1A</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T50">GLIB_000:50</a></i>;<br/>
<a NAME="E10:76"/><i><font color="Green">E64</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2A</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G1A</font>
 
<b>by </b><i>, <a class="ref" href="glib_002.html#D2" target="_self">Def2</a>, <a class="ref" href="glib_000.html#T50">GLIB_000:50</a></i>;<br/>
<a NAME="E11:76"/>
<font color="Maroon">G2</font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">G1</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T90">GLIB_000:90</a></i>;<br/>
<a NAME="E12:76"/><b>then </b>
<font color="Maroon">G2A</font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T46">GLIB_000:46</a></i>;<br/>
<a NAME="E13:76"/><b>then </b>
<font color="Maroon">G1A</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G2A</font>
 
<b>by </b><i><a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, , <a class="ref" href="glib_000.html#T89">GLIB_000:89</a></i>;<br/>
<b>hence </b><a NAME="E14:76"/>
<font color="Maroon">G2</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a>  <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">G2A</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a> 
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, </i>;<br/>


</div>
